Newform orbit 243.2.c.c

• Label: 243.2.c.c
• Level: $243$
• Weight: $2$
• Character orbit: 243.c
• Analytic conductor: $1.940$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94036476912\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 2\nu ) / 2 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{3} + 4\nu ) / 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1 + \beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

82.1

1.22474	+	0.707107i
−1.22474	−	0.707107i
1.22474	−	0.707107i
−1.22474	+	0.707107i

−1.22474

−

2.12132i

0

−2.00000

+

3.46410i

1.22474

−

2.12132i

0

−1.00000

−

1.73205i

4.89898

0

−6.00000

82.2

1.22474

+

2.12132i

0

−2.00000

+

3.46410i

−1.22474

+

2.12132i

0

−1.00000

−

1.73205i

−4.89898

0

−6.00000

163.1

−1.22474

+

2.12132i

0

−2.00000

−

3.46410i

1.22474

+

2.12132i

0

−1.00000

+

1.73205i

4.89898

0

−6.00000

163.2

1.22474

−

2.12132i

0

−2.00000

−

3.46410i

−1.22474

−

2.12132i

0

−1.00000

+

1.73205i

−4.89898

0

−6.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.2.c.c		4
3.b	odd	2	1	inner	243.2.c.c		4
9.c	even	3	1		243.2.a.d	✓	2
9.c	even	3	1	inner	243.2.c.c		4
9.d	odd	6	1		243.2.a.d	✓	2
9.d	odd	6	1	inner	243.2.c.c		4
27.e	even	9	6		729.2.e.p		12
27.f	odd	18	6		729.2.e.p		12
36.f	odd	6	1		3888.2.a.z		2
36.h	even	6	1		3888.2.a.z		2
45.h	odd	6	1		6075.2.a.bn		2
45.j	even	6	1		6075.2.a.bn		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.2.a.d	✓	2	9.c	even	3	1
243.2.a.d	✓	2	9.d	odd	6	1
243.2.c.c		4	1.a	even	1	1	trivial
243.2.c.c		4	3.b	odd	2	1	inner
243.2.c.c		4	9.c	even	3	1	inner
243.2.c.c		4	9.d	odd	6	1	inner
729.2.e.p		12	27.e	even	9	6
729.2.e.p		12	27.f	odd	18	6
3888.2.a.z		2	36.f	odd	6	1
3888.2.a.z		2	36.h	even	6	1
6075.2.a.bn		2	45.h	odd	6	1
6075.2.a.bn		2	45.j	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(243, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 36 \)

\( T_{7}^{2} + 2T_{7} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 6T^{2} + 36 \)
$3$	\( T^{4} \)
$5$	\( T^{4} + 6T^{2} + 36 \)
$7$	\( (T^{2} + 2 T + 4)^{2} \)
$11$	\( T^{4} + 6T^{2} + 36 \)